Consider \(n\)  independent random variables \(X^{(i)}\in [a,b]\)  with its mean \({E}(\bar{X} )= {E}(\frac{1}{n}\sum X^{(i)} )\) .
The Hoeffding theorem [1]} gives a simple bound on the deviation of the observed average \(\bar{X}\)  after \(n\)  trails from the expected value \({E}(\bar{X} )\)
\(\textrm {P}\Big (\bar{X} - t \ge {E}(\bar{X}) \Big )\le \exp \left(-\frac{2 n t^2}{(b-a)^2}\right).\) 
